Existence of Solutions for a Weighted Integral equation with Negative Power

Gang Ding; Haiyang He

doi:10.3934/cpaa.2022157

Article Contents

2023, Volume 22, Issue 2: 441-458. Doi: 10.3934/cpaa.2022157

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Existence of Solutions for a Weighted Integral equation with Negative Power

Gang Ding and
Haiyang He^,

Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China

^* Corresponding author: Haiyang He
^* Corresponding author: Haiyang He

Received: June 2022

Revised: September 2022

Early access: October 2022

Published: February 2023

The corresponding author are supported by Hunan Provincial Natural Science Foundation of China (No:2022JJ30366), and Research Foundation of Education of Hunan Province, China(No:20A293)

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Abstract

In this paper, our purpose is to study the following weighted integral equation with negative power

$ \begin{equation} f^{q-1}(x) = \int_{\Omega}\dfrac {|x|^{\alpha}|y|^{\alpha}f(y)}{|x-y|^{N-\mu}} \ dy \ +\lambda\int_{\Omega}\dfrac {f(y)}{|x-y|^{N-\mu-1}} \ dy \ , f\geq0, x\in\overline{\Omega}\;(0.1) \end{equation} $

where $ \Omega $ is a bounded domain with smooth boundary in $ \mathbb{R}^N $, $ 0\in \Omega $, $ \mu>N $, $ \alpha<0 $ and $ \lambda $ is a real parameter. We prove that there exists a positive solution of problem (0.1) if $ 0<q\leq q_\mu = \frac{2N}{N+\mu} $ with a suitable $ \lambda $.

Keywords:

Nonlinear Integral equation,
existence solutions,
reversed Hardy-Littlewood-Sobolev inequality,
weight.

Mathematics Subject Classification: Primary: 45G10, 35J60.

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